The MLSVGD proposed in this work builds on SVGD introduced by [1]} and further theoretically analyzed in [2]}; extended to consider Newton directions [3]}; exploiting geometry [4]}, and other acceleration techniques [5]}.
A key building block for us will be recent advances on understanding the convergence properties of SVGD in the infinite particle (mean-field) regime. The work [2]}, [7]} shows the mean-field limit. The work [8]} shows non-asymptotic results. Further, the work [9]} establishes exponential convergence under certain situations in the mean-field limit that motivates some of our assumptions.
Another key building block is relating discretization error of the forward model at level \(\ell \)  to divergence of the corresponding posterior distributions \(\pi ^{(\ell )}\)  with respect to the intractable target \(\pi \) , where we build on results by [10]} and an inequality involving the Kullback-Leibler (KL) divergence introduced by [11]}.
